Dr. Alexandra I. Cristea

http://www.dcs.warwick.ac.uk/~acristea/

CS 319: Theory of Databases

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Content1. Generalities DB2. Temporal Data3. Integrity constraints (FD revisited)4. Relational Algebra (revisited)5. Query optimisation6. Tuple calculus7. Domain calculus8. Query equivalence9. LLJ, DP and applications10. The Askew Wall11. Datalog

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… previous

FD revisited; proofs with FD with definition & counter-example

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FD Part 2: Proving with FDs:

• Proving with Armstrong axioms

• (non)Redundancy of FDs

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Armstrong’s Axioms

• Axioms for reasoning about FD’s

F1: reflexivity reflexivity if Y X then X Y

F2: augmentationaugmentation if X Y then XZ YZ

F3: transitivitytransitivity if X Y and Y Z then X Z

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Theorems• Additional rules derived from axioms:

F4. UnionUnion

if A B and A C, then A BC

F5. DecompositionDecomposition

if A BC, then A B and A C

• Prove them!

A B

C

AB

C

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Union Rule

if A B and A C, then A BC

• Let A B and A C

• A B, augument (F2) with A: A AB

• A C, augument (F2) with B: AB BC

• A AB and AB BC, apply transitivity (F3): A BC q.e.d.

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Decomposition Rule

if A BC, then A B and A C• Let A BC• BBC, apply reflexivity (F1) : BC B • A BC and BC B, apply transitivity

(F3): A B• Idem for A C q.e.d.

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Rules hold vs redundant?

• Armstrong Rules hold – but are they all necessary?

• Can we leave some out?– How do we check this?

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RedundancyDEF: An inference rule inf in a set of

inference rules Rules for a certain type of constraint C is redundant (superfluous) when for all sets F of constraints of type C it holds that:

F+{Rules –{inf}} = F+

{Rules}

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F+, F*

• F+ = {fd | F |= fd} closure of F

• F* = {fd | F |- fd} cover of F

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Exercises

1. Show that Armstrong’s inference rules for FDs (F1-3) are not redundant.

2. Show that Rules = {F1, F2, F3, F4} is redundant.

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Hint (Ex. 1)

• Show with the help of an example that, if one of the three axioms is omitted, the remaining set of functional dependencies is not complete.

• Take therefore an appropriate set of constraints and compute with the help of Rules – {inf} all possible consequences. Show then that there is another consequence to be computed with the help of inf.

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Solution• We start from a relation scheme R and an arbitrary

legal instance r(R). Let , and be sets of attributes (headers), so that Attr(R), Attr(R) and Attr(R). We have the following axioms:

• F1: (Reflexivity) Let be valid (holds). Then we also have →.

• F2: (Augmentation) Let → be valid. Then we also have →.

• F3: (Transitivity) Let → and → be valid. Then we also have →.

• Now we omit in turn one of the axioms.– Why in turn?– Why not just one?

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Case 1: F1 is not superfluous:

• Let Attr(R) = {X} and F = . Because F is empty, neither F2 nor F3 can be used to deduce new fds. Therefore, F+ = F = .

• From F1 we could however deduce that X X is valid, which is not present in the above set.

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Case 3: F2 is not superfluous:

• Let R = {X, Y} and F = {X Y}.• With the help of F1 and F3 we deduce:• F+ = { , X X, Y Y, X , Y

, XY XY, XY Y, XY X, XY }

• However, with X Y and with the help of F2 we can infer that X XY is valid, which is not present in the above set.

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Case 3: F3 is not superfluous:• Let R = {X, Y, Z} and F = {X Y, Y Z }.F+ = {

XYZ XYZ, XY XY, YZ YZ, X Y, Y Z,XYZ XY, XY X, YZ Y,X XY, Y YZ,XYZ XZ, XY Y, YZ Z,XY Y, XY XZ,

XYZ YZ, XY , YZ , XZ YZ, YZ Z,XYZ X, XZ XZ, X X, X , Y Y, Y , Z Z, Z XYZ Y,XYZ Z, XYZ , XZ , }

• With the help of F3 we can also infer X Z, which is not in F+.

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How do we show something is redundant (superfluous)?

• Show that it is inferable from the other axioms

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F4 is superfluous:• F4 (union rule) : Let → and → be valid. • Then → is also valid.• We show now that F = {F1, F2, F3, F4} is redundant is by, e.g., inferring F4

from the other three. • By using augumentation, from → we deduce that also → is valid

(augmentation with ).• By using augumentation, from → we deduce that also → is valid

(augmentation with ).• By using transitivity, from → and →, we deduce that also → is

valid.

• Note that to prove that a set of rules (axioms) is redundant we can use normal calculus; however, to prove that a set of rules is not redundant, we need to know the meaning of the rules.

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Summary

• We have learned how to prove fds based on the Armstrong axioms – and also why & when it’s ok to do so

• We have learned how to prove that a set of axioms is redundant or not

• We have learned that the Armstrong axioms are not redundant

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… to follow

Constrains revisited: Soundness and Completeness of Armstrong Axioms